# Adaptive Hurst-Sensitive Active Queue Management

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. State of the Art

## 3. Hurst Estimation Methods

- $H\in (0;0.5)$—the process is negatively correlated, which means that the Long-Range Dependence does not occur;
- $H=0.5$—the process is uncorrelated;
- $H\in (0.5;1)$—the process is positively correlated, which means that the LRD occurs.

## 4. Adaptive AQM

## 5. Selection of the AQM Parameters with the Use of Neural Networks

#### Adaptive Neuron AQM

## 6. Results

#### 6.1. Fluid Flow Analysis

- ${W}_{i}$ is the expected TCP congestion window size (in packets) for the i-th flow. It defines a number of packets that may be sent without waiting for the acknowledgements of the reception of previous packets;
- ${R}_{i}$ is the round-trip time, ${R}_{i}=q/C+{T}_{p}$, the sum $\sum \frac{{W}_{i}}{{R}_{i}}$ denotes the total input flow to the congestion router;
- q is queue length (in packets);
- C is link capacity (packets/time unit), the constant output flow of the router;
- ${T}_{p}$ is propagation delay;
- N is the number of TCP sessions passing through the router;
- p is the packet drop probability.

- transmission capacity of AQM router: $C=0.075$;
- propagation delay for i-th flow: ${T}_{{p}_{i}}=2$;
- starting time for i-th flow (TCP and UDP);
- the number of packets sent by i-th flow (TCP and UDP).

- $Mi{n}_{th}=10$;
- $Ma{x}_{th}=15$;
- buffer size (measured in packets) $=20$;
- ${P}_{max}=0.1$;
- eight parameter $w=0.007$.

#### 6.2. Simulation

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Router average queue length values, Fluid Flow approximation, ARED Hurst-insensitive 1 TCP stream (

**left**,

**top**), ARED Hurst-sensitive 1 TCP stream (

**right**,

**top**) and ANRED Hurst-insensitive 1 TCP stream (

**left**,

**bottom**), ANRED with Hurst-sensitive 1 TCP stream (

**right**,

**bottom**).

**Figure 4.**Router average queue length values, Fluid Flow approximation, $ANP{I}^{\alpha}$ Hurst-insensitive 1 TCP stream (

**left**,

**top**), $ANP{I}^{\alpha}$ Hurst-sensitive 1 TCP stream (

**right**,

**top**) and $ANP{I}^{\alpha}{D}^{\beta}$ Hurst-insensitive 1 TCP stream (

**left**,

**bottom**), $ANP{I}^{\alpha}{D}^{\beta}$ Hurst-sensitive 1 TCP stream (

**right**,

**bottom**).

**Figure 6.**Router queue length values, $\mu $ = 0.25, ARED Hurst-insensitive algorithm, $\alpha =0.5$, H = 0.5 (

**left**,

**top**), H = 0.9 (

**right**,

**top**) and ARED Hurst-sensitive algorithm, $\alpha =0.5$, H = 0.5 (

**left**,

**bottom**), H = 0.9 (

**right**,

**bottom**).

**Figure 7.**Router queue length values, $\mu $ = 0.25, ANRED Hurst-insensitive algorithm, $\alpha =0.5$, H = 0.5 (

**left**,

**top**), H = 0.9 (

**right**,

**top**) and ANRED Hurst-sensitive algorithm, $\alpha =0.5$, $\mu $ = 0.25, H = 0.5 (

**left**,

**bottom**), H = 0.9 (

**right**,

**bottom**).

**Figure 8.**Queue lengths, $\mu $ = 0.25, $P{I}^{\alpha}$ Hurst-insensitive algorithm, $\alpha =0.5$, H = 0.5 (

**left**,

**top**), H = 0.9 (

**right**,

**top**) and $P{I}^{\alpha}$ Hurst-sensitive algorithm, $\alpha =0.5$, H = 0.5 (

**left**,

**bottom**), H = 0.9 (

**right**,

**bottom**).

**Figure 9.**Parameters evolution, $\mu $ = 0.25, $P{I}^{\alpha}$ Hurst-insensitive algorithm, $\alpha =0.5$, H = 0.5 (

**left**,

**top**), H=0.9 (

**right**,

**top**) and $P{I}^{\alpha}$ Hurst-sensitive algorithm, $\alpha =0.5$, H = 0.5 (

**left**,

**bottom**), H = 0.9 (

**right**,

**bottom**).

**Figure 10.**Queue length values, $\mu $ = 0.25, $P{I}^{\alpha}{D}^{\beta}$ Hurst-insensitive algorithm, $\alpha =0.5$, H = 0.5 (

**left**,

**top**), H = 0.9 (

**right**,

**top**) and $P{I}^{\alpha}{D}^{\beta}$ Hurst-sensitive algorithm, $\alpha =0.5$, H = 0.5 (

**left**,

**bottom**), H = 0.9 (

**right**,

**bottom**).

H | Method 1 | Method 2 |
---|---|---|

0.5 | 0.4975 | 0.4975 |

0.6 | 0.5918 | 0.5918 |

0.7 | 0.7124 | 0.7124 |

0.8 | 0.8098 | 0.8098 |

0.9 | 0.9108 | 0.9108 |

n | Method 1 | Method 2 (ver. 1) | Method 2 (ver. 2) |
---|---|---|---|

${2}^{10}$ | 0.000992 | 0.000995 | 0.000009 |

${2}^{12}$ | 0.003968 | 0.004454 | 0.000010 |

${2}^{14}$ | 0.015376 | 0.018848 | 0.000011 |

${2}^{16}$ | 0.062462 | 0.076383 | 0.000013 |

${2}^{18}$ | 0.262380 | 0.311451 | 0.000014 |

Hurst | Mean | Lost | No. of Dropped Packets | Delay | ||
---|---|---|---|---|---|---|

Queue Length | AQM | Queue | Average | Min–Max | ||

0.5 | 22.93 | 0.49% | 19,261 | 266 | 0.092 | 2.04 · 10${}^{-2}$–0.18 |

0.6 | 23.05 | 0.49% | 19,270 | 341 | 0.093 | 3.12 · 10${}^{-3}$–0.19 |

0.7 | 23.55 | 0.54% | 22,950 | 605 | 0.089 | 3.14 · 10${}^{-4}$–0.18 |

0.8 | 23.31 | 0.57% | 25,943 | 919 | 0.086 | 1.80 · 10${}^{-4}$–0.20 |

0.9 | 22.56 | 0.65% | 34,040 | 717 | 0.081 | 1.46 · 10${}^{-6}$–0.18 |

Hurst | Mean | Lost | No. of Dropped Packets | Delay | ||
---|---|---|---|---|---|---|

Queue Length | AQM | Queue | Average | Min–Max | ||

0.5 | 22.30 | 0.49% | 19,391 | 290 | 0.091 | 6.94 · 10${}^{-3}$–0.18 |

0.6 | 20.99 | 0.50% | 19,306 | 359 | 0.082 | 5.49 · 10${}^{-3}$–0.18 |

0.7 | 19.92 | 0.54% | 23,275 | 364 | 0.073 | 3.24 · 10${}^{-4}$–0.18 |

0.8 | 19.17 | 0.58% | 26,873 | 268 | 0.072 | 2.21 · 10${}^{-5}$– 0.21 |

0.9 | 19.51 | 0.65% | 34,873 | 47 | 0.068 | 2.52 · 10${}^{-6}$–0.16 |

Hurst | Mean | Lost | No. of Dropped Packets | Delay | ||
---|---|---|---|---|---|---|

Queue Length | AQM | Queue | Average | Min–Max | ||

0.5 | 18.24 | 0.50% | 19,498 | 297 | 0.073 | 7.47 · 10${}^{-3}$–0.16 |

0.6 | 18.14 | 0.50% | 19,179 | 590 | 0.073 | 9.25 · 10${}^{-3}$–0.18 |

0.7 | 18.55 | 0.55% | 22,865 | 944 | 0.070 | 1.33 · 10${}^{-4}$–0.18 |

0.8 | 18.64 | 0.58% | 25,591 | 1409 | 0.068 | 8.04 · 10${}^{-5}$–0.16 |

0.9 | 19.19 | 0.65% | 33,786 | 1272 | 0.067 | 2.05 · 10${}^{-5}$–0.18 |

Hurst | Mean | Lost | No. of Dropped Packets | Delay | ||
---|---|---|---|---|---|---|

Queue Length | AQM | Queue | Average | Min–Max | ||

0.5 | 18.37 | 0.50% | 19,290 | 476 | 0.073 | 5.39 · 10${}^{-2}$–0.17 |

0.6 | 18.06 | 0.49% | 18,790 | 586 | 0.073 | 1.27 · 10${}^{-2}$–0.18 |

0.7 | 18.45 | 0.55% | 22,964 | 916 | 0.070 | 2.08 · 10${}^{-4}$–0.18 |

0.8 | 18.09 | 0.58% | 26,124 | 1247 | 0.069 | 1.46 · 10${}^{-5}$–0.19 |

0.9 | 18.49 | 0.65% | 34,195 | 934 | 0.069 | 6.22 · 10${}^{-5}$–0.19 |

Hurst | Mean | Lost | No. of Dropped Packets | Delay | ||
---|---|---|---|---|---|---|

Queue Length | AQM | Queue | Average | Min–Max | ||

0.5 | 14.40 | 0.49% | 18942 | 655 | 0.048 | 3.77 · 10${}^{-3}$–0.12 |

0.6 | 14.45 | 0.49% | 18655 | 705 | 0.048 | 4.38 · 10${}^{-4}$–0.11 |

0.7 | 14.57 | 0.54% | 22628 | 1051 | 0.047 | 9.87 · 10${}^{-6}$–0.12 |

0.8 | 14.55 | 0.58% | 25867 | 1147 | 0.044 | 1.33 · 10${}^{-5}$–0.10 |

0.9 | 14.41 | 0.66% | 34215 | 1027 | 0.044 | 1.62 · 10${}^{-6}$–0.13 |

Hurst | Mean | Lost | No. of Dropped Packets | Delay | ||
---|---|---|---|---|---|---|

Queue Length | AQM | Queue | Average | Min–Max | ||

0.5 | 10.26 | 0.50% | 19622 | 89 | 0.039 | 1.68 · 10${}^{-4}$–0.10 |

0.6 | 10.27 | 0.49% | 19417 | 81 | 0.039 | 1.26 · 10${}^{-4}$–0.10 |

0.7 | 10.24 | 0.54% | 23562 | 52 | 0.038 | 4.94 · 10${}^{-5}$–0.10 |

0.8 | 10.31 | 0.59% | 27245 | 316 | 0.037 | 1.10 · 10${}^{-5}$–0.10 |

0.9 | 10.27 | 0.66% | 35145 | 189 | 0.037 | 4.76 · 10${}^{-6}$–0.10 |

Hurst | Mean | Lost | No. of Dropped Packets | Delay | ||
---|---|---|---|---|---|---|

Queue Length | AQM | Queue | Average | Min–Max | ||

0.5 | 14.48 | 0.50% | 19,163 | 772 | 0.049 | 5.29 · 10${}^{-3}$–0.13 |

0.6 | 14.52 | 0.50% | 18,935 | 755 | 0.049 | 1.30 · 10${}^{-4}$–0.11 |

0.7 | 14.59 | 0.54% | 22,590 | 1041 | 0.048 | 4.76 · 10${}^{-5}$–0.15 |

0.8 | 14.57 | 0.58% | 25,870 | 1196 | 0.045 | 1.11 · 10${}^{-5}$–0.11 |

0.9 | 14.32 | 0.65% | 34,000 | 867 | 0.043 | 2.51 · 10${}^{-5}$–0.13 |

Hurst | Mean | Lost | No. of Dropped Packets | Delay | ||
---|---|---|---|---|---|---|

Queue Length | AQM | Queue | Average | Min–Max | ||

0.5 | 10.28 | 0.50% | 19,806 | 62 | 0.040 | 1.19 · 10${}^{-5}$–0.12 |

0.6 | 10.27 | 0.51% | 20,096 | 55 | 0.039 | 1.97 · 10${}^{-4}$–0.10 |

0.7 | 10.25 | 0.54% | 23,688 | 70 | 0.038 | 2.23 · 10${}^{-4}$–0.10 |

0.8 | 10.33 | 0.59% | 27,212 | 355 | 0.037 | 2.04 · 10${}^{-5}$–0.10 |

0.9 | 10.23 | 0.65% | 35,078 | 125 | 0.037 | 6.33 · 10${}^{-6}$–0.12 |

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**MDPI and ACS Style**

Marek, D.; Szyguła, J.; Domański, A.; Domańska, J.; Filus, K.; Szczygieł, M.
Adaptive Hurst-Sensitive Active Queue Management. *Entropy* **2022**, *24*, 418.
https://doi.org/10.3390/e24030418

**AMA Style**

Marek D, Szyguła J, Domański A, Domańska J, Filus K, Szczygieł M.
Adaptive Hurst-Sensitive Active Queue Management. *Entropy*. 2022; 24(3):418.
https://doi.org/10.3390/e24030418

**Chicago/Turabian Style**

Marek, Dariusz, Jakub Szyguła, Adam Domański, Joanna Domańska, Katarzyna Filus, and Marta Szczygieł.
2022. "Adaptive Hurst-Sensitive Active Queue Management" *Entropy* 24, no. 3: 418.
https://doi.org/10.3390/e24030418